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Electron transport through single and multiple quantum dots: The formation of a one-dimensional bandstructure
Superlattices and Microstructures,
197
Vol. 10, No. 2, 1991
ELECTRON TRANSPORT THROUGH SINGLE AND MULTIPLE QUANTUM DOTS: THE FORMATION OF A ONBDIMENSIONAL BANDSIB UCTUBE B. J. van Wees, L.P. Kouwenhoven, A. van der Enden, and C.J.P.M. Hat-mans Faculty of Applied Physics, De@ University of Technology P.O.Box 5046,2600 GA De@, The Netherlands C. E. Timmering Philips Research Laboratories, 5600 JA Eimfhoven, The Netherlands (Received
13 August 1990)
We describe transport experiments performed on ballistic submicron devices which are defined in the two dimensional electron gas of GaAs/AlGaAs heterostructures by means of metallic gates. Conductance measurements on single quantum dots reveal the formation of magnetically induced zero-dimensional (OD) states. In an array of 15 coupled quantum dots, these OD-states develop into a 1D bandstructute.
1. Introduction In the field of semiconductor Dhvsics, the electron transnort through ballistic conductors h&attained much attentioit in the last few yearsl. With a variety of fabrication techniaues it has become possible to make impurity-free submi&on conductors in which the electron transport is denoted as ballistic. Moreover, when the dimensions of the conductor are comparable to the De Broglie wavelength LP of the electrons at the Fermi energy, quantum phenomena become important. In this case, the electron transport is called quantum ballistic, which will be the main subject of this paper. One of the first results obtained in the quantum ballistic transport regime is the conductance quantization of a quantum point contact (QPC)*. A QPC is a short ballistic constriction with variable width defined in a two dimensional electron gas (ZDEG). On changing the width of the constriction it was found that at zero-magnetic field the conductance changes in quantized steps of 2e*/h. This result is a demonstration of transport through 1D subbands which are formed due to the lateral confinement in the constriction. Changing the width moves the Fermi energy relative to the 1D subbands and each time the number of occupied subbands changes by one, the conductance changes by 2&h. An annealine theoretical formalism to describe the conduct&e qiantization and many other transport nhenomena is the Landauer-Btlttiker formalism3. The 1D subbands are viewed here as 1D current channels, each current channel cotmibuting T 2e*/h to the conductance. A fully transmitted current channel (which has a transmission probability T=l) from the soume reservoir to the sink reservoir then contributes the quantized value 2&h to the conductance. The Landauer-Bllttiker formalism has also become important to describe transport in the quantum Hall
0749-6036/91/060197
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$02.00/O
effect (QHE) regime. When the Fermi energy is located between two Landau levels in the bulk of the ZDEG, the relevant electron states for transport (those at the Fermi energy) are located at the boundaries of the sample4. These so-called edge states or edge channels follow equipotential lines at the boundaries of the 2DEG, and electrons which move in edge channels at opposite boundaries have opposite velocities. The transport through edge channels is 1D and each spin-resolved channel contributes e*/h to the Hall conductance. If NL Landau levels are occupied in the bulk, also NJ_ edge channels carry a net current, which yields a total Hall conductance of GB-NL e%. It was pointed out by Bilttiker3 that due to the separation of leftand right-going edge channels at opposite boundaries of the sample, backscattering involves scattering from one sample edge to the opposite edge. When the Fermi energy is between two Landau levels in the bulk, this backscatterring is sumrressed because of the absence of available extended electrbn states. Experiments by Komiyama et al.5, van Wees et al.6, and Alnhenaar et a1.7 have shown that forward scattering beiween edge channels at the same boundary can also be. extremely small. On short distances of order pm this forward scattering can he vhtually absent, for which reason edge channels can be treated as independent 1D current channels. Recent theontica18 and experimental9 work has suggested that the edge channel picture may be valid to describe transport in the fractional quantum Hall effect regime as well. The one-dimensionality of the transm in combination with the absence of scattering. make edge channels an ideal starting point to study the OD-regime. In the subsequent sections the fonnrtion of magnetical.ly induced OD-states in a single quantum dotlo, and the formation of a 1D band structurtinapcriodicmrayofcoupledquantumdotsttwill be discussed.
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and Microstructures,
Vol. 10, No. 2, 7991
second edge channel is only partially transmitted through the two QPCs with probabilities TA and Tg. Inside the dot this edge channel forms a 1D loop, which encloses a magnetic flux @=BA, with A the enclosed area. The contribution of the second edge channel to the conductance is equal to T2 e2/h, where the transmission probability T2 of the confined edge channel loop is given by: TATB
T2 =
1 - 2 4 RARB
Figure 1. Schematic layout of the quantum dot with diameter of 1.5pm and two 300nm wide quantum point contacts. The electron flow in edge channels is shown when a high magnetic field B is applied. A 1D loop is formed which encloses a flux CD,when an edge channel is only partially transmitted by both quantum point contacts.
2. Transport
through a single quantum
dot.
The device is defined in the 2DEG of a high mobility GaAs/AlGaAs heterosnucture. The 2DEG has an electron density of 2.3.1015m-2 and a transport mean free path of 9pm. On top of the heterostructure two pairs of metallic gates A and B are fabricated by means of electron-beam lithography (see fig.1). Application of a negative voltage of -0.2V on the gates, depletes the electron gas underneath the gates and creates a quantum dot in the 2DEG with a diameter of l.Spm. To allow electron transport, a connection from the two wide 2DEG regions to the quantum dot is arranged by two 300nm wide QPCs. The narrow channel between the two gate pairs A and B is already pinched-off at this gate voltage. Applying the gate voltage to only one gate pair and zero voltage to the other, defines a single QPC in the 2DEG. In this way the transport properties of the individual QPCs can be measured and be conipared to that of the comgete device. The conductance measurements are performed at a temperature of IOmK, in a high magnetic field (>lT), by biasing a small current through the sample (clnA) and by measuring the resulting voltage between the two wide 2DEG ngions. In ref.12 we have used the gate geometry of fig.1 to study how the conductance GD of the two QPCs in series is related to the conductances GA and GB of the individual QPCs. At zero-magnetic field, we found a simple Ohmic addition I/GD=I/GA+~/GB~~. However, when a magnetic field is applied, an enhancement of the series conductance was observed, and above 1T the series conductance became equal to the smallest conductance of one of the individual QPCs: Gn=min(GA,GR). ._ _. In this case, scattering urocesses b;etween>dge channels are completely abse:i and the transport is called adiabatic (i.e. electrons travel through the device with conservation of quantum-edge channelnumber). The important consequence of adiabatic transport is that edge channels can be treated as independent 1D current channels. The location of two independent edge channels in the quantum dot is schematically shown in fig. 1 The first edge channel is completely transmitted through the total device and therefore contributes ez/h to the conductance. The
COS(V)
+
RARB
(1)
with R=l-T and the phase v is defined by v=2x@l@o where @o=h/e is the fluxquantum. The total conductance GD is just the sum of all the contributions and is given for M completely transmitted edge channels by: e2
GD=T;(M+T~)
(2)
Eqs. 1 and 2 imply that when the enclosed flux Q, is changed the conductance oscillates with a period equal to the flixquantum @o. Ea.1 is known to describe a 1D interferometer, which con&ts out of two barriers with transmission probabilities TA and Tg placed at a distance a. For that case, tbe phase v is given by v=2x(2a/hF). When v is an integer times 27~ constructive interference occurs between the two barriers which results in maximal total transmission T2 (when TA=TB and v=integep2x, it follows from eq.1 that T2=1). This fact that the total transmission can be larger than the transmissions of the individual barriers, is known as resonant tunneling. By changing the phase v, one can tune the interference from constructive to destructive (or the transmission T2 from maximal to minimal) just like in an optical interferometer. In terms of energy, the electron states between the two barriers split up due to the finite length a, resulting in single electronic OD-states. When the Fermi energy coincides with a OD-state the transmission T2 becomes resonant. The observation of OD-states at B=O has been reported by Smith et al.14 who were able to vary the distance a between the two barriers, and by Hirayama and Sakuls who varied the Fermi energy. In our approach using edge channels, the OD-states are formed when the enclosed flux 0 is an integer times the fluxquantum 0016. The advantage of this approach is that scattering in the dot is absent and that the confined edge channels are 1D by themselves. Fig& shows GD versus B at constant Vg=-?.35V in the transition region between the second to the third plateau, which corresponds to the complete transmission of two (soin resolved) edge channels and the partial transmission &the third ch&mei. Large oscillations &e seen between the two plateaus with an amplitude modulation up to 40% of e2/h. Fig.Zb displays the oscillations on an expanded scale showing their regularity. The period B. of the oscillations smoothly varies from Bo=2.5mT at B=2ST to Bo=2.8mT at B=2.7T. This period cannot be deduced directly from the enclosed area A, because the location of the edge channels at the boundary also deoends on BlO. In fig.2~ the region of low tran&sion or weak coupling is plotted. Here the conductance is 2e2/h (the third edge channel is completely reflected here), except when the Fermi energy coincides with the energy of a OD-state. The narrow conductance
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Figure 3. Conductance oscillations as a function of gate voltage for a fixed magnetic field B=2ST. In (a) the voltage on both gate pairs is varied (period=lmV) and in (b) the gate voltage on one pair is kept fixed at -0.35V and varied on the other gate pair (period=2mV).
3. Transport L__.__
2.55
~
2.7
MAGNETIC
2.75
FIELD
(T)
Figure 2. (a) Transition from the second to the third plateau of the dot conductance GD, showing large oscillations with amplitudes up to 40% of e*/h. (b) Enlarged oscillations from (a) showing their regularity (period Bo=2.5mT ). (c) Region of low transmissions of the third edge channel (TA,TB=O). The discrete conductance peaks demonstrate resonant transmission through OD-states.
peaks demonstrate the resonant transmission through ODstates. It is shown elsewhere10 that a calculation of GD from eq. 1 with the measured conductances of the individual QPCs substituted for the transmissions TA and TB, gives good agreement with the measured GD shown in fig.2. In fig.3 it is demonstrated that the phase v can also be varied by changing the area of the dot. Here GD is measured as a function of gate voltage for a fixed B=2ST. In fig.3a the voltage is varied on both gates (which affects the total area of the dot) and the oscillation period is 1mV. In fig.3b the voltage on one gate pair is kept fixed and varied on the other (which affects only half of the dot area) and, as expected, the oscillation period is now 2mV. The above measurements demonstrate that the conductance of a quantum dot in a high magnetic field can be strongly modulated by changing the magnetic flux enclosed by the current carrying states. If the Fermi energy coincides with the energy of a OD-state, the transmission through the dot is at resonance. The rounded oscillations for large trans’missions TA and TB and the sharp peaks for low TA and Tg illustrate the effect of coupling of the discrete OD-states with the continuum of states in the ID edge channel in the wide 2DEG regions outside the dot, which can be shown to be in accordance with eq. 1.
through
multiple
quantum
dots
One of the basic principles of solid state theory is that when atoms are put together in a periodic array, the discrete atomic states develop into collective states: the band structure. A similar band structure can also be expected when quantum dots, in which OD-states are formed as described in the previous section, are placed in a periodic array. We have realized such a device acting as an artificial 1D crystal which will be discussed in this section. The gate geometry of the 1D crystal device is shown schematically in the inset of fig.6. The ungated 2DEG of this sample has an electron density of 2.7.1015m-* and a transport mean free path of lOl.tm. A voltage Vgl on the first gate defines in total 16 fingers in the 2DEG at a period of 200nm. The effect of making the voltage Vg2 more negative is to increase the depletion region around the second gate. This reduces the area of each dot and lowers the Fermi energy in the conducting regions. The potential landscape in the 2DEG presumably resembles a smooth periodic electrostatic confinement, with the asymmetric saddle-shaped maxima defined by the fingers. The transport through a finite periodic 1D potential has been calculated by different authors using different methods*‘. We have extended eq.1 to a recursive formula describing the transmission probability TN of N barriers in series”: TN=~N~N*;
tN =
ttN_1 1 -rrN_l
eXp(iv)
(3)
In fig.4 TN is plotted as a function of phase for 16 barriers and for three different transmissions T of the individual barriers. Due to the coupling (expressed by T), the OD states develop into bands characterized by a high transmission TN. They are separated by gaps with low TN. The bands consist of a number of discrete states equal to the number of quantum dots N-l. As can be seen also in fig.4, weak coupling results in large gaps (tight binding regime), while strong coupling yields small gaps (nearly free electron regime).
Superlattices
Figure 4. Transmission TN versus phase v/2x:of a periodic 1D chain of 16 barriers, for different single barrier transmissions T.
B=2T
GATE
VOLTAGE-?
(V)
Figure 5. Conductance as a function of gate voltage V 2 on the second gate at a fixed magnetic field of 2 T an a for different values of gate voltage V 1 on the first gate. Corresponding peaks are marked. Tf!e curves have been offset for clarity.
The conductance measurements in this section are rformed at 1OmK and at a fued magnetic field of B=2T. r lg.5 shows tbe first quantum Hall plateau for several fixed values Vat and changing the voltage Vg2 on the second gate. Large oscillations are seen below the first plateau (i.e. G
and Microstructures,
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Figure 6. Conductance as a function of gate voltage Vg2 on the second gate at 2 T and Vat = -0.45V on the fust gate. The inset schematically shows the gate geometry; the dashed lines indicate the depletion regions in the 2DEG.
and the smaller oscillations to the discrete states, which form the bands. Going vertically through fig.4, one can also see that when the transmissions T are varied relatively fast compared to the phase v, the Fermi energy moves through the band structure. This situation corresponds to the large oscillations below the first plateau, where T changes from 0 (at pinch-off) to 1. The effect of irregularities on the bandstructure can also be seen in fig.5. The trace of Vgt= -0.46V shows 16 smaller oscillations between the marks “*‘I and “+“, and in the curve of V,t= -0.47V the gap marked by “+” is not seen. Note also the smaller oscillations on the left of the mark “*” indicating another band, although clearly less regular. The 16 oscillations between two gaps cannot be explained by the number of quantum dots, not even when they are considered to be. unequal. An additional scatterer somewhere in the device could be the cause- of this extra oscillation. The disappearance of a gap is presumably due to irregularities, which introduces mixing between bands. This also follows from further calculations11 of the transmission TN from eq.3 for N=16, in which irregularities are introduced by variations in the transmissions T of the individual barriers. With one barrier chosen in the middle of the crystal of which T deviates by a few percent, the calculations show that the smaller oscillations group together in pairs of two, very similar as seen in some parts of fig.5. 5. Conclusions The state of the art of submicron technology has reached the level where one is able to study fundamental solid state phenomena as for instance 1D transport, OD-states, and artificial band structures. These phenomena can be visualized in transport measurements in a very simple way, and can be described by physically transparent theories as the Landauer-Btittiker formalism. Acknowledgement - We thank M. van Eijck, F.W.J. Hekking, W. Kool. J.R. Kraayeveld, N.C. van der Vaart for valuable discussions, and the Stichting F.O.M. for financial support.
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and Microstructures,
Vol. 10, No. 2, 1991
References 1. For reviews see: M. Reed and W.P. Kirk, eds., Nanostructure Physics and Fabrication (Academic Press, New York, 1989j; S.P. Beamont and C.M. Sotomayorand Engineering of I- and OTorres, eds.,Science Dimensional Semiconductors (Plenum, London, 1989); J.M. Chamberlain, L. Eaves, and J.C. Portal, eds., Electronic Properties of Multilayersand Low-Dimensional Semiconducror Structures (Plenum, London, 1990). 2. B.J. van Wees et al., Phys. Rev. Lett. 60, 848.(1988); D.A. Wharam et al., J. Phys. C 21, L209 (1988). 3. for a review see: M. Bilttiker, volume of Semiconductors and Semimetals, Nanostructured Systems edited by M.A. Reed (Academic Press, Orlando, Florida, 1990). 4. RI. Halperin, Phys. Rev. B 25,2185 (1982). 5. S. Komiyama et al., Phys. Rev B 40, 12566 (1989). 6. B.J. van Wees et al., Phys. Rev. L&t. 62, 1181 (1989). 7. R.W. Alnhenaar et al., Phvs. Rev. Len. 64,677 (1990). 8. C.W.J. Beenakker, Phys: Rev. Lea. 64.216 (1990); A.H. MacDonald, Phys. Rev. Lett. 64, 220 (1990); A.M. Chang, Solid State Commun. 74, 871 (1990). 9. A.M. Chang and J.E. Cunnigham, Solid State Commun. 72, 651 (1989); L.P. Kouwenhoven et al., Phys. Rev. Len. 64,685 (1990). 10. see also: B.J. van Wees et al., Phys. Rev. Lett. 62, 2523 (1989): L.P. Kouwenhoven et al., Proc. of the
201 workshop on Science and Engineering of 1- and Odimensional Semiconductors, edited by C.M. SotomayorTorres and S.P. Beaumont (Plenum Press, 1990); L.P. Kouwenhoven et al., Proc. of the EP2DS8, Surface Science 229,290 (1990). 11. see also: L.P. Kouwenhoven et al., Phys. Rev. Lea. 65, 361 (1990); M. van Eijck et al., Proc. of LTI9 (Physica B, to be published). 12. L.P. Kouwenhoven et al., Phys. Rev. B 40, 8083 (1989); A similar structure has been studied experimentally by: P.C. Main, et al. Phys. Rev. B 40, 10033 (1989). and theoretically by: L.J. Glazman and M. Johnson, J. Phys. Condens. Matter 1,5547 (1989). 13. An enhancement of the series conductance above Ohmic addition at B=O was found by: D.A. Wharam et al., J. Phys. C 21, L887 (1988). 14. C.G. Smith et al., J. Phys. Condens. Matter 1, 9035 (1989). 15. Y. Himyama and T. Saku, Solid State Comm. 73, 113 (1990). 16. A theoretical study of a 2D dot to which narrow leads are attached is given by: U. Sivan. Y. Imry, and C. Hartzstein, Phys. Rev. B 39, 1242 (1989); U. Sivan, and Y. Imry, Phys. Rev. Len. 61,lOOl (1988). 17. R. Tsu and L. Es&i, ADDS. Phvs. Lett. 22. 562 (1973); D.J. Vezzetti and h4.9: Cahay, J. Phys. d. 19, L53 (1986): H.W. Lee. A. Zvsnarski. and P. Kerr. Am. J. Phys: 57, ‘729 (1989); S.E: Ulloa, ‘E. Castano ‘and G. Kirczenow, Phys. Rev. B 41.12350 (1990).
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Vol. 10, No. 2, 1991
ELECTRON TRANSPORT THROUGH SINGLE AND MULTIPLE QUANTUM DOTS: THE FORMATION OF A ONBDIMENSIONAL BANDSIB UCTUBE B. J. van Wees, L.P. Kouwenhoven, A. van der Enden, and C.J.P.M. Hat-mans Faculty of Applied Physics, De@ University of Technology P.O.Box 5046,2600 GA De@, The Netherlands C. E. Timmering Philips Research Laboratories, 5600 JA Eimfhoven, The Netherlands (Received
13 August 1990)
We describe transport experiments performed on ballistic submicron devices which are defined in the two dimensional electron gas of GaAs/AlGaAs heterostructures by means of metallic gates. Conductance measurements on single quantum dots reveal the formation of magnetically induced zero-dimensional (OD) states. In an array of 15 coupled quantum dots, these OD-states develop into a 1D bandstructute.
1. Introduction In the field of semiconductor Dhvsics, the electron transnort through ballistic conductors h&attained much attentioit in the last few yearsl. With a variety of fabrication techniaues it has become possible to make impurity-free submi&on conductors in which the electron transport is denoted as ballistic. Moreover, when the dimensions of the conductor are comparable to the De Broglie wavelength LP of the electrons at the Fermi energy, quantum phenomena become important. In this case, the electron transport is called quantum ballistic, which will be the main subject of this paper. One of the first results obtained in the quantum ballistic transport regime is the conductance quantization of a quantum point contact (QPC)*. A QPC is a short ballistic constriction with variable width defined in a two dimensional electron gas (ZDEG). On changing the width of the constriction it was found that at zero-magnetic field the conductance changes in quantized steps of 2e*/h. This result is a demonstration of transport through 1D subbands which are formed due to the lateral confinement in the constriction. Changing the width moves the Fermi energy relative to the 1D subbands and each time the number of occupied subbands changes by one, the conductance changes by 2&h. An annealine theoretical formalism to describe the conduct&e qiantization and many other transport nhenomena is the Landauer-Btlttiker formalism3. The 1D subbands are viewed here as 1D current channels, each current channel cotmibuting T 2e*/h to the conductance. A fully transmitted current channel (which has a transmission probability T=l) from the soume reservoir to the sink reservoir then contributes the quantized value 2&h to the conductance. The Landauer-Bllttiker formalism has also become important to describe transport in the quantum Hall
0749-6036/91/060197
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$02.00/O
effect (QHE) regime. When the Fermi energy is located between two Landau levels in the bulk of the ZDEG, the relevant electron states for transport (those at the Fermi energy) are located at the boundaries of the sample4. These so-called edge states or edge channels follow equipotential lines at the boundaries of the 2DEG, and electrons which move in edge channels at opposite boundaries have opposite velocities. The transport through edge channels is 1D and each spin-resolved channel contributes e*/h to the Hall conductance. If NL Landau levels are occupied in the bulk, also NJ_ edge channels carry a net current, which yields a total Hall conductance of GB-NL e%. It was pointed out by Bilttiker3 that due to the separation of leftand right-going edge channels at opposite boundaries of the sample, backscattering involves scattering from one sample edge to the opposite edge. When the Fermi energy is between two Landau levels in the bulk, this backscatterring is sumrressed because of the absence of available extended electrbn states. Experiments by Komiyama et al.5, van Wees et al.6, and Alnhenaar et a1.7 have shown that forward scattering beiween edge channels at the same boundary can also be. extremely small. On short distances of order pm this forward scattering can he vhtually absent, for which reason edge channels can be treated as independent 1D current channels. Recent theontica18 and experimental9 work has suggested that the edge channel picture may be valid to describe transport in the fractional quantum Hall effect regime as well. The one-dimensionality of the transm in combination with the absence of scattering. make edge channels an ideal starting point to study the OD-regime. In the subsequent sections the fonnrtion of magnetical.ly induced OD-states in a single quantum dotlo, and the formation of a 1D band structurtinapcriodicmrayofcoupledquantumdotsttwill be discussed.
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and Microstructures,
Vol. 10, No. 2, 7991
second edge channel is only partially transmitted through the two QPCs with probabilities TA and Tg. Inside the dot this edge channel forms a 1D loop, which encloses a magnetic flux @=BA, with A the enclosed area. The contribution of the second edge channel to the conductance is equal to T2 e2/h, where the transmission probability T2 of the confined edge channel loop is given by: TATB
T2 =
1 - 2 4 RARB
Figure 1. Schematic layout of the quantum dot with diameter of 1.5pm and two 300nm wide quantum point contacts. The electron flow in edge channels is shown when a high magnetic field B is applied. A 1D loop is formed which encloses a flux CD,when an edge channel is only partially transmitted by both quantum point contacts.
2. Transport
through a single quantum
dot.
The device is defined in the 2DEG of a high mobility GaAs/AlGaAs heterosnucture. The 2DEG has an electron density of 2.3.1015m-2 and a transport mean free path of 9pm. On top of the heterostructure two pairs of metallic gates A and B are fabricated by means of electron-beam lithography (see fig.1). Application of a negative voltage of -0.2V on the gates, depletes the electron gas underneath the gates and creates a quantum dot in the 2DEG with a diameter of l.Spm. To allow electron transport, a connection from the two wide 2DEG regions to the quantum dot is arranged by two 300nm wide QPCs. The narrow channel between the two gate pairs A and B is already pinched-off at this gate voltage. Applying the gate voltage to only one gate pair and zero voltage to the other, defines a single QPC in the 2DEG. In this way the transport properties of the individual QPCs can be measured and be conipared to that of the comgete device. The conductance measurements are performed at a temperature of IOmK, in a high magnetic field (>lT), by biasing a small current through the sample (clnA) and by measuring the resulting voltage between the two wide 2DEG ngions. In ref.12 we have used the gate geometry of fig.1 to study how the conductance GD of the two QPCs in series is related to the conductances GA and GB of the individual QPCs. At zero-magnetic field, we found a simple Ohmic addition I/GD=I/GA+~/GB~~. However, when a magnetic field is applied, an enhancement of the series conductance was observed, and above 1T the series conductance became equal to the smallest conductance of one of the individual QPCs: Gn=min(GA,GR). ._ _. In this case, scattering urocesses b;etween>dge channels are completely abse:i and the transport is called adiabatic (i.e. electrons travel through the device with conservation of quantum-edge channelnumber). The important consequence of adiabatic transport is that edge channels can be treated as independent 1D current channels. The location of two independent edge channels in the quantum dot is schematically shown in fig. 1 The first edge channel is completely transmitted through the total device and therefore contributes ez/h to the conductance. The
COS(V)
+
RARB
(1)
with R=l-T and the phase v is defined by v=2x@l@o where @o=h/e is the fluxquantum. The total conductance GD is just the sum of all the contributions and is given for M completely transmitted edge channels by: e2
GD=T;(M+T~)
(2)
Eqs. 1 and 2 imply that when the enclosed flux Q, is changed the conductance oscillates with a period equal to the flixquantum @o. Ea.1 is known to describe a 1D interferometer, which con&ts out of two barriers with transmission probabilities TA and Tg placed at a distance a. For that case, tbe phase v is given by v=2x(2a/hF). When v is an integer times 27~ constructive interference occurs between the two barriers which results in maximal total transmission T2 (when TA=TB and v=integep2x, it follows from eq.1 that T2=1). This fact that the total transmission can be larger than the transmissions of the individual barriers, is known as resonant tunneling. By changing the phase v, one can tune the interference from constructive to destructive (or the transmission T2 from maximal to minimal) just like in an optical interferometer. In terms of energy, the electron states between the two barriers split up due to the finite length a, resulting in single electronic OD-states. When the Fermi energy coincides with a OD-state the transmission T2 becomes resonant. The observation of OD-states at B=O has been reported by Smith et al.14 who were able to vary the distance a between the two barriers, and by Hirayama and Sakuls who varied the Fermi energy. In our approach using edge channels, the OD-states are formed when the enclosed flux 0 is an integer times the fluxquantum 0016. The advantage of this approach is that scattering in the dot is absent and that the confined edge channels are 1D by themselves. Fig& shows GD versus B at constant Vg=-?.35V in the transition region between the second to the third plateau, which corresponds to the complete transmission of two (soin resolved) edge channels and the partial transmission &the third ch&mei. Large oscillations &e seen between the two plateaus with an amplitude modulation up to 40% of e2/h. Fig.Zb displays the oscillations on an expanded scale showing their regularity. The period B. of the oscillations smoothly varies from Bo=2.5mT at B=2ST to Bo=2.8mT at B=2.7T. This period cannot be deduced directly from the enclosed area A, because the location of the edge channels at the boundary also deoends on BlO. In fig.2~ the region of low tran&sion or weak coupling is plotted. Here the conductance is 2e2/h (the third edge channel is completely reflected here), except when the Fermi energy coincides with the energy of a OD-state. The narrow conductance
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Vol. 10, No. 2, 199 1
Figure 3. Conductance oscillations as a function of gate voltage for a fixed magnetic field B=2ST. In (a) the voltage on both gate pairs is varied (period=lmV) and in (b) the gate voltage on one pair is kept fixed at -0.35V and varied on the other gate pair (period=2mV).
3. Transport L__.__
2.55
~
2.7
MAGNETIC
2.75
FIELD
(T)
Figure 2. (a) Transition from the second to the third plateau of the dot conductance GD, showing large oscillations with amplitudes up to 40% of e*/h. (b) Enlarged oscillations from (a) showing their regularity (period Bo=2.5mT ). (c) Region of low transmissions of the third edge channel (TA,TB=O). The discrete conductance peaks demonstrate resonant transmission through OD-states.
peaks demonstrate the resonant transmission through ODstates. It is shown elsewhere10 that a calculation of GD from eq. 1 with the measured conductances of the individual QPCs substituted for the transmissions TA and TB, gives good agreement with the measured GD shown in fig.2. In fig.3 it is demonstrated that the phase v can also be varied by changing the area of the dot. Here GD is measured as a function of gate voltage for a fixed B=2ST. In fig.3a the voltage is varied on both gates (which affects the total area of the dot) and the oscillation period is 1mV. In fig.3b the voltage on one gate pair is kept fixed and varied on the other (which affects only half of the dot area) and, as expected, the oscillation period is now 2mV. The above measurements demonstrate that the conductance of a quantum dot in a high magnetic field can be strongly modulated by changing the magnetic flux enclosed by the current carrying states. If the Fermi energy coincides with the energy of a OD-state, the transmission through the dot is at resonance. The rounded oscillations for large trans’missions TA and TB and the sharp peaks for low TA and Tg illustrate the effect of coupling of the discrete OD-states with the continuum of states in the ID edge channel in the wide 2DEG regions outside the dot, which can be shown to be in accordance with eq. 1.
through
multiple
quantum
dots
One of the basic principles of solid state theory is that when atoms are put together in a periodic array, the discrete atomic states develop into collective states: the band structure. A similar band structure can also be expected when quantum dots, in which OD-states are formed as described in the previous section, are placed in a periodic array. We have realized such a device acting as an artificial 1D crystal which will be discussed in this section. The gate geometry of the 1D crystal device is shown schematically in the inset of fig.6. The ungated 2DEG of this sample has an electron density of 2.7.1015m-* and a transport mean free path of lOl.tm. A voltage Vgl on the first gate defines in total 16 fingers in the 2DEG at a period of 200nm. The effect of making the voltage Vg2 more negative is to increase the depletion region around the second gate. This reduces the area of each dot and lowers the Fermi energy in the conducting regions. The potential landscape in the 2DEG presumably resembles a smooth periodic electrostatic confinement, with the asymmetric saddle-shaped maxima defined by the fingers. The transport through a finite periodic 1D potential has been calculated by different authors using different methods*‘. We have extended eq.1 to a recursive formula describing the transmission probability TN of N barriers in series”: TN=~N~N*;
tN =
ttN_1 1 -rrN_l
eXp(iv)
(3)
In fig.4 TN is plotted as a function of phase for 16 barriers and for three different transmissions T of the individual barriers. Due to the coupling (expressed by T), the OD states develop into bands characterized by a high transmission TN. They are separated by gaps with low TN. The bands consist of a number of discrete states equal to the number of quantum dots N-l. As can be seen also in fig.4, weak coupling results in large gaps (tight binding regime), while strong coupling yields small gaps (nearly free electron regime).
Superlattices
Figure 4. Transmission TN versus phase v/2x:of a periodic 1D chain of 16 barriers, for different single barrier transmissions T.
B=2T
GATE
VOLTAGE-?
(V)
Figure 5. Conductance as a function of gate voltage V 2 on the second gate at a fixed magnetic field of 2 T an a for different values of gate voltage V 1 on the first gate. Corresponding peaks are marked. Tf!e curves have been offset for clarity.
The conductance measurements in this section are rformed at 1OmK and at a fued magnetic field of B=2T. r lg.5 shows tbe first quantum Hall plateau for several fixed values Vat and changing the voltage Vg2 on the second gate. Large oscillations are seen below the first plateau (i.e. G
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Figure 6. Conductance as a function of gate voltage Vg2 on the second gate at 2 T and Vat = -0.45V on the fust gate. The inset schematically shows the gate geometry; the dashed lines indicate the depletion regions in the 2DEG.
and the smaller oscillations to the discrete states, which form the bands. Going vertically through fig.4, one can also see that when the transmissions T are varied relatively fast compared to the phase v, the Fermi energy moves through the band structure. This situation corresponds to the large oscillations below the first plateau, where T changes from 0 (at pinch-off) to 1. The effect of irregularities on the bandstructure can also be seen in fig.5. The trace of Vgt= -0.46V shows 16 smaller oscillations between the marks “*‘I and “+“, and in the curve of V,t= -0.47V the gap marked by “+” is not seen. Note also the smaller oscillations on the left of the mark “*” indicating another band, although clearly less regular. The 16 oscillations between two gaps cannot be explained by the number of quantum dots, not even when they are considered to be. unequal. An additional scatterer somewhere in the device could be the cause- of this extra oscillation. The disappearance of a gap is presumably due to irregularities, which introduces mixing between bands. This also follows from further calculations11 of the transmission TN from eq.3 for N=16, in which irregularities are introduced by variations in the transmissions T of the individual barriers. With one barrier chosen in the middle of the crystal of which T deviates by a few percent, the calculations show that the smaller oscillations group together in pairs of two, very similar as seen in some parts of fig.5. 5. Conclusions The state of the art of submicron technology has reached the level where one is able to study fundamental solid state phenomena as for instance 1D transport, OD-states, and artificial band structures. These phenomena can be visualized in transport measurements in a very simple way, and can be described by physically transparent theories as the Landauer-Btittiker formalism. Acknowledgement - We thank M. van Eijck, F.W.J. Hekking, W. Kool. J.R. Kraayeveld, N.C. van der Vaart for valuable discussions, and the Stichting F.O.M. for financial support.
SWerlattices
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References 1. For reviews see: M. Reed and W.P. Kirk, eds., Nanostructure Physics and Fabrication (Academic Press, New York, 1989j; S.P. Beamont and C.M. Sotomayorand Engineering of I- and OTorres, eds.,Science Dimensional Semiconductors (Plenum, London, 1989); J.M. Chamberlain, L. Eaves, and J.C. Portal, eds., Electronic Properties of Multilayersand Low-Dimensional Semiconducror Structures (Plenum, London, 1990). 2. B.J. van Wees et al., Phys. Rev. Lett. 60, 848.(1988); D.A. Wharam et al., J. Phys. C 21, L209 (1988). 3. for a review see: M. Bilttiker, volume of Semiconductors and Semimetals, Nanostructured Systems edited by M.A. Reed (Academic Press, Orlando, Florida, 1990). 4. RI. Halperin, Phys. Rev. B 25,2185 (1982). 5. S. Komiyama et al., Phys. Rev B 40, 12566 (1989). 6. B.J. van Wees et al., Phys. Rev. L&t. 62, 1181 (1989). 7. R.W. Alnhenaar et al., Phvs. Rev. Len. 64,677 (1990). 8. C.W.J. Beenakker, Phys: Rev. Lea. 64.216 (1990); A.H. MacDonald, Phys. Rev. Lett. 64, 220 (1990); A.M. Chang, Solid State Commun. 74, 871 (1990). 9. A.M. Chang and J.E. Cunnigham, Solid State Commun. 72, 651 (1989); L.P. Kouwenhoven et al., Phys. Rev. Len. 64,685 (1990). 10. see also: B.J. van Wees et al., Phys. Rev. Lett. 62, 2523 (1989): L.P. Kouwenhoven et al., Proc. of the
201 workshop on Science and Engineering of 1- and Odimensional Semiconductors, edited by C.M. SotomayorTorres and S.P. Beaumont (Plenum Press, 1990); L.P. Kouwenhoven et al., Proc. of the EP2DS8, Surface Science 229,290 (1990). 11. see also: L.P. Kouwenhoven et al., Phys. Rev. Lea. 65, 361 (1990); M. van Eijck et al., Proc. of LTI9 (Physica B, to be published). 12. L.P. Kouwenhoven et al., Phys. Rev. B 40, 8083 (1989); A similar structure has been studied experimentally by: P.C. Main, et al. Phys. Rev. B 40, 10033 (1989). and theoretically by: L.J. Glazman and M. Johnson, J. Phys. Condens. Matter 1,5547 (1989). 13. An enhancement of the series conductance above Ohmic addition at B=O was found by: D.A. Wharam et al., J. Phys. C 21, L887 (1988). 14. C.G. Smith et al., J. Phys. Condens. Matter 1, 9035 (1989). 15. Y. Himyama and T. Saku, Solid State Comm. 73, 113 (1990). 16. A theoretical study of a 2D dot to which narrow leads are attached is given by: U. Sivan. Y. Imry, and C. Hartzstein, Phys. Rev. B 39, 1242 (1989); U. Sivan, and Y. Imry, Phys. Rev. Len. 61,lOOl (1988). 17. R. Tsu and L. Es&i, ADDS. Phvs. Lett. 22. 562 (1973); D.J. Vezzetti and h4.9: Cahay, J. Phys. d. 19, L53 (1986): H.W. Lee. A. Zvsnarski. and P. Kerr. Am. J. Phys: 57, ‘729 (1989); S.E: Ulloa, ‘E. Castano ‘and G. Kirczenow, Phys. Rev. B 41.12350 (1990).